Numerical solution of variational inequalities for pricing Asian options by higher order Lagrange–Galerkin methods

Asian options prices can be modelled in the Black–Scholes framework leading to two-factor models depending on the asset price, the average of the asset price and the time. They can also involve inequality constraints, as in the case of Amerasian options, leading to variational inequalities (VI). In the first section, we completely describe the pricing model for fixed-strike Eurasian and Amerasian options and list some properties satisfied by the option value function. Then, since no solutions in closed form are known, we deal with the numerical solution of the above problems proposing a general methodology: an iterative algorithm for the VI, combined with higher order Lagrange–Galerkin methods for partial differential equations. Finally, numerical results are shown.